existing between the two systems. These rules of transformation are given by the Einstein-Lorentz transformations.[63]
Now these transformations are, as we have said, more restricted than the conformal transformations; and this lesser generality of the Einstein-Lorentz transformations has, as a consequence, the further restriction of the conditions of invariance of the mathematical expression mentioned previously. Not only will this expression have a zero value for all Galilean frames when it has a zero value for one particular Galilean frame, but in addition, if it does not happen to have a zero value in one frame but has some definite non-vanishing numerical value, it will still maintain this same definite non-vanishing value in all other Galilean frames. In other words, Einstein’s premises are represented mathematically by the invariance of the total value of
for all Galilean frames, regardless of whether this value happens to be zero or non-vanishing.
The deep significance of this condition of invariance was first noted by Minkowski, and it led, as we shall explain in the next chapter, to the discovery of four-dimensional space-time.
CHAPTER XVIII
THE DISCOVERY OF SPACE-TIME
THE discovery of this invariant
whose value we shall designate by